t-Statistics Under Fat-Tailed Returns

A strategy's backtest produces a sequence of daily returns, and a researcher computes a t-statistic to test whether the average return is significantly different from zero. This test relies on the central limit theorem — the claim that the average of enough independent draws is approximately normally distributed, regardless of the shape of the individual draws. That's true eventually. But daily and even monthly financial returns are notoriously fat-tailed — occasional extreme moves far more frequent than a normal distribution would predict — and fat tails slow down exactly how quickly "eventually" arrives. A t-test that looks perfectly rigorous can be built on an averaging-out assumption that hasn't actually kicked in yet.

An analogy: averaging heights versus averaging wealth

If you average the heights of 30 random people, the average settles down quickly, close to the true population mean — heights have no extreme outliers, so 30 is plenty for it to behave normally. If you instead average the wealth of 30 random people, one billionaire can single-handedly dominate the average, making it wildly different from a different random 30 — extreme values mean the average doesn't stabilize nearly as fast, and formulas assuming quick stabilization badly understate how much it can still swing. Daily trading returns sit much closer to the wealth example than the height one.

The idea, one symbol at a time

The central limit theorem guarantees that $\overline{X}$, the sample mean of $n$ i.i.d. returns with finite mean $\mu$ and variance $\sigma^2$, converges toward $N(\mu, \sigma^2/n)$ as $n \to \infty$ — this convergence justifies computing $t = (\overline{X} - 0)/(s/\sqrt{n})$ and comparing it to normal critical values. The catch is "as $n \to \infty$": the rate of convergence depends on the distribution's higher moments (skewness, kurtosis). For light-tailed data, convergence is fast — often good by $n=30$–50. For fat-tailed data (high excess kurtosis, or infinite-variance Pareto-type models), it can require thousands of observations before the normal approximation is trustworthy, with the true finite-sample distribution of $\overline{X}$ visibly fatter-tailed than the CLT implies at any practical sample size.

A useful diagnostic is excess kurtosis $\kappa$ (kurtosis minus 3, the normal distribution's value): $\kappa=0$ for normal data, and daily equity returns commonly show $\kappa$ from 3 to 10 or higher. Larger $\kappa$ means slower convergence and a less trustworthy t-statistic at any given $n$ — the spikier the return distribution looks against a bell curve, the more data is needed before the usual critical value of 1.96 is actually accurate.

Worked example 1: a strategy with near-normal daily returns

A market-neutral strategy has daily returns with excess kurtosis $\kappa \approx 0.5$ (close to normal), mean daily return $0.02\%$, standard deviation $0.5\%$, over $n = 250$ trading days:

$$t = \frac{0.0002}{0.005/\sqrt{250}} = \frac{0.0002}{0.000316} \approx 0.63 .$$

Not significant, but here the CLT has almost certainly converged well by $n=250$ given the near-normal kurtosis, so the t-statistic and its associated p-value can be trusted at face value.

Worked example 2: the same numbers, but fat-tailed returns

Now suppose an options-selling-style strategy has the same sample mean (0.02%) and standard deviation (0.5%) over the same $n=250$ days, producing the identical $t \approx 0.63$ — but its daily returns have excess kurtosis $\kappa \approx 15$ (fat-tailed, driven by rare large losing days). Simulation studies at this kurtosis level show the actual sampling distribution of $\overline{X}$ at $n=250$ still has meaningfully heavier tails than the CLT's normal approximation predicts — the true probability of seeing $|t| \ge 0.63$ by chance is higher than the normal table says, and a nominal 5% test might have a true false-positive rate closer to 8–10%. Identical t-statistics, very different actual reliability.

What this means in practice

Before trusting a backtest's t-statistic, check the return series' kurtosis (or eyeball a histogram against a normal curve) — a t-statistic on visibly fat-tailed daily returns, especially with modest sample sizes (a few hundred days, a typical one- or two-year backtest), deserves real skepticism even if it clears 1.96. Standard remedies: bootstrap the empirical distribution for a data-driven critical value, use robust or heavy-tailed test statistics, or simply demand a much larger sample before declaring significance for strategies with known fat-tailed risk (short-volatility, credit, insurance-like payoffs).